Some remarks on the quantum geometry

In this paper we study the Schrödinger equation and the integrability problem at the quantum level from the geometric quantization point of view.     

Some remarks on the quantum double

We recall the abstract theory of Hopf algebra bicrossproducts and double cross products due to the author. We use it to develop some less-well known results about the quantum double as a twisting, as an extension and asq-Lorentz group.     

Some remarks about the localization of states in a quantum field theory

For the case of a field theory with a nuclear space of test functions (for instance, the space of strongly decreasing test functions) compact sets of states are constructed; these correspond to sets of localized states. Only such states are considered which are elements of a fixed subspace of the entire Hilbert space. This subspace belongs to them-point functions of order less than a certain fixed 2n.Supported in part by the Air Force Office of Scientific Research under Grant AF 500–64On leave of absence from the Max Planck-Institut für Physik und Astrophysik, München.     

Some remarks on classical representations of quantum mechanics

It is shown that, to a certain extent, the statistical framework of Hilbert-space quantum mechanics can be reformulated in classical terms.     

Some remarks on the “null plane development” of a relativistic quantum field theory

We give conditions for the existence of field operators on so-called null planes and discuss some consequences of the necessary restriction of the test function space, concerning Haag's theorem and the possibility of unitary mappings intertwining between free fields of different masses. In the last section we discuss conditions under which a unitary representation of the dilatations in the null plane gives rise to a unitary representation of the dilatations in Minkowski space.     

Some remarks on local operators in quantum electrodynamics

We assume the existence of a conserved current which generates locally gauge transformations of first kind. We are working in a local quantum Field Theory, where the fields are defined on a vector space where indefinite metric is allowed. We show that the Maxwell equations are not consistent with the above assumptions and the vectors obtained by applying local charged operators on the vacuum cannot describe physical states. Moreover we show that, if charged fields have non-trivial expectation value on the physical states, the vector space must contain vectors with negative norm. We discuss the relation between the local formulation of QED and a formulation in terms of physical states. As an example we study the transition from Gupta-Bleuler free QED to the Coulomb-gauge formulation.     

Some remarks on the time-dependent Thomas-Fermi theory

We investigate the Thomas Fermi limit (?=0) of the density operator in phase space for a system of noninteracting fermions evolving from the ground state in a time dependent potential. The semiclassical calculations for model situations in one spatial dimension are compared with the solution of the time dependent Schrödinger equation. The role of time dependent invariants and the relation to the hydrodynamical formulation of quantum mechanics is pointed out.     

Causality in local quantum field theory: Some general results

General asymptotic causality properties of chronologicalN-point functions and, in massive theories, ofN-particle collision amplitudes, are derived from locality and the spectral condition. Results include specified rates of exponential fall-off, with simple and direct physical content, for large non-causal separations of points or particles in Minkowski space-time depending on values of the energy-momenta and on the mass spectrum. Relevant mathematical results on rates of exponential fall-off of generalized Fourier transforms outside their microsupports are given.Laboratoire de la Direction des Sciences de la Matière du Commissariat à l'Energie Atomique     

Some remarks on a model of supersymmetric quantum mechanics

We develop a recently proposed model within supersymmetric quantum mechanics that puts a group structure on the creation and annihilation operators. We apply the scheme to a variety of quantum mechanical problems and work out a two-term energy recursion equation when the overall group structure isU(1, 1).     

Some remarks on radiation gauges in general relativity

We prove that it is possible to specify the shift vector field for a given foliation in such a way as to provide automatically the transverse-traceless part of the second fundamental form K associated with the slices of the foliation. We interpret this particular specification as providing a simple way for characterizing the dynamical degrees of freedom of the gravitational field. The resulting gauge is not the minimal distortion gauge devised by Smarr and York. In numerical relativity it permits the maintenance in time of the difference equations associated with the divergence constraint.     

Some remarks on the determination of quantum states by measurements

The problem of state determination of quantum systems by the probability distributions of some observables is considered. In particular, we review a question already asked by W. Pauli, namely, the determination of pure states of spinless particles by the distributions of position and momentum. In this context we give a new example of two wave functions differing by a piecewise constant phase having the same position and momentum distributions. ThePauli problem is investigated also under incorporation of special types of the Hamiltonian. Moreover, in case of spin-1 systems with three-dimensional Hilbert space, it is shown that the probabilities for the values of six suitably chosen spin components determine their state.     

Some remarks on the energy-momentum tensor in general relativity

In general relativity, the energy-momentum tensor of a classical tensor field can be constructed by varying the action of the field with respect to the background metric. This paper suggests an alternative interpretation of the construction which also makes sense for spinor fields, and which gives some insight into the locality of energy-momentum operators in generally covariant quantum field theory.     

On the general covariance and strong equivalence principles in quantum general relativity

The various physical aspects of the general relativistic principles of covariance and strong equivalence are discussed, and their mathematical formulations are analyzed. All these aspects are shown to be present in classical general relativity, although no contemporary formulation of canonical or covariant quantum gravity has succeeded to incorporate them all. This has, in part, motivated the recent introduction of a geometro-stochastic framework for quantum general relativity, in which the classical frame bundles that underlie the formulation of parallel transport in classical general relativity are replaced by quantum frame bundles. It is shown that quantum frames can take over the role played by complete sets of observables in conventional quantum theory, so that they can mediate the natural transference of the general covariance and the strong equivalence principles from the classical to the quantum general relativistic regime. This results in a geometrostochastic mode of quantum propagation in general relativistic quantum bundles, which is mathematically implemented by path integration methods based on parallel transport along horizontal lifts of geodesics for the vacuum expectation values of a quantum gravitational field in a quantum spacetime supermanifold. The covariance features of this field are embedded in a quantum gravitational supergroup, which incorporates Poincaré as well as diffeomorphism invariance, and resolves the issue of time in quantum gravity.     

Some remarks concerning the equivalence principle of general relativity with respect to quantum mechanical one-particle state

We discuss a recent experiment with interfering neutron beams in the gravitational field of the earth, and show how the explanation of the interference pattern can be obtained by an application of the equivalence principle to the one-particle states. The calculation is carried out in the coordinate system where the gravitational field has been transformed away.     

Some remarks concerning stability for nonstationary quantum systems

The problem of characterizing stability and instability for general nonstationary quantum systems is investigated. Some characterizations are reported and some elementary properties of a topological characterization are established. Then, it is proven, by considering a simple example, that there are nonperiodic driven systems whose orbits are neither precompact nor leave on average any compact set. Autocorrelation measures are computed and the possible roles of the generalizes quasienergy operator and energy growth are briefly discussed.     

Some remarks on the Liouville approach to two-dimensional quantum gravity

We show the equivalence of tow-dimensional quantum gravity interacting with a total number of scalar and Dirac-fermion degrees of freedom less of 26 and quantum Liouvill field theory. An analysis of the two-dimensional effective geometry, as well as of the conditions of non-triviality of the resulting quantum theory, is also included. A general procedure to get the complete one-loop effective action is suggested at the end.     

Some remarks on nonlocal field theory and space-time quantization

From the point of view that the charge and mass of an electron is of dynamical origin and quantization of charge in units ofe is related to the space-time quantization as developed in an earlier paper, we here show that it is possible to consider that the internal space within the elementary domain of the quantized space-time world is not governed by Lorentz invariance. This helps us to develop a consistent theory of nonlocal fields for extended particles where the infinite mass degeneracy is avoided. Moreover, this ensures the convergence of nonlocal field theories and suggests that massless particles like photons and neutrinos, though they may be taken to be of extended structure, will appear only as point particles in the physical world. In this picture, Lorentz invariance appears to be a consequence of the distribution of matter and energy in the Universe, and this may be taken to be another interpretation of Mach's principle.     

Some remarks on the relativistic theory of pursuit and evasion

In the present paper we have formulated the problem of the non-closed differential game of non-zero sum. As an example, the problem of pursuit and evasion in relativistic theories has been analysed. The main results of the paper are as follows: (a) Equilibrium conditions of the non-closed game of non-zero sum in Euclidian-Minkowski (EM) and Riemann-Einstein (RE) spaces. (b) Canonical form of the differential equations for the equilibrium points of the game (upper and lower values of the game), min-max and max-min strategies). (c) Interpretation of the basic notions and rules of the differential game for two persons in EM and RE spaces.     

Some remarks on -invariant Fedosov star products and quantum momentum mappings

In these notes we consider a slightly generalized Fedosov star product * on a symplectic manifold (M,ω), emanating from the fibrewise Weyl product and the triple (,Ω,s) consisting of a symplectic torsion free connection on M, a formal series ΩνZ²_dR(M)[[ν]] of closed two-forms on M, and a certain formal series s of symmetric contravariant tensor fields on M. We prove necessary and sufficient conditions for certain classical symmetries to become symmetries of the star product, only sufficient conditions having been published in special cases when this letter was written (note, however, the different proofs in [S. Gutt, J. Rawnsley, Natural star products on symplectic manifolds and quantum moment maps, 2003. math.SG/0304498 v1]). For a given symplectic vector field X on M, it is well known that (= is a sufficient condition for the Lie derivative to be a derivation of *. We prove that these conditions are in fact necessary ones, also providing a very simple proof for their being sufficient. Moreover, we prove a criterion that has first been presented by Gutt [S. Gutt, Star products and group actions, Contribution to the Bayrischzell Workshop, April 26–29, 2002] (see also [S. Gutt, J. Rawnsley, Natural star products on symplectic manifolds and quantum moment maps, 2003. math.SG/0304498 v1] for a different proof) and which specifies a necessary and sufficient condition for to be a quasi-inner derivation. The statement that this condition is a sufficient one dates back to Kravchenko [O. Kravchenko, Compos. Math. 123 (2000) 131]. Applying our results, we find necessary and sufficient criteria for a Fedosov star product to be -invariant and to admit a quantum Hamiltonian. Finally, supposing the existence of a quantum Hamiltonian, we present a cohomological condition on Ω that is equivalent to the existence of a quantum momentum mapping. In particular, our results show that the existence of a classical momentum mapping in general does not imply the existence of a quantum momentum mapping and thus give a negative answer to Xu’s question posed in [P. Xu, Commun. Math. Phys. 197 (1998) 167].